We implement the pseudo likelihood formulations of [1]} and [2]} by replacing the exact Bernoulli likelihood for the observed sample under our main method of Section  with approximate likelihoods for the underlying population estimated on the sample. Both methods parameterize only \(\pi _{ci}\)  for convenience units and use the inclusion probabilities for the reference sample as a plug-in. Let vector \(\mathbf {\theta }_{c} = \left(\mathbf {\gamma }_{x,c},(\mathbf {\beta }_{ck})_{k=1}^{K}\right)\)  denote the parameters in the non-linear logistic regression model for \(\pi _{ci}(\mathbf {x}_{i},\mathbf {\theta }_{c})\) . [1]} specify the following pseudo log-likelihood,
\(\ell (\mathbf {\theta }_{c}) = \mathop {\sum }_{i \in S_{c}}\log \left(\frac{\pi _{ci}(\mathbf {x}_{i},\mathbf {\theta }_{c})}{1-\pi _{ci}(\mathbf {x}_{i},\mathbf {\theta }_{c})}\right) + \mathop {\sum }_{i \in S_{r}}d_{ri}\log \left(1-\pi _{ci}\left(\mathbf {x}_{i},\mathbf {\theta }_{c}\right)\right),\) 
